Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

Abdon Atangana, Badr Alkahtani
2015 Entropy  
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of
more » ... led-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order. 4440 applications of information theory can be found in [1] [2] [3] [4] [5] . The field of fractional order derivatives has attracted the attention of many researchers in all branches of sciences and engineering. In recent years, many field of sciences and technology have used fractional order derivatives to model many real world problems in their respective fields, as it has been revealed that these fractional order derivatives are very efficient in describing such problems [6] [7] [8] [9] [10] [11] [12] . It is no wonder therefore why many researchers in the field of fractional calculus have devoted their attention to proposing new fractional order derivatives [13] [14] [15] [16] [17] [18] [19] . These derivative definitions range from the well-known Riemann-Liouville derivative to the newly proposed one known as the Caputo-Fabrizio derivative. It is very important to note that most of these definitions are based on the convolution. The definitions proposed by Riemann-Liouville and the first Caputo version has the weakness that their kernel had singularity. Since the kernel is used to describe the memory effect of the system, it is clear that with this weakness, these two derivatives cannot accurately describe the full effect of the memory. To further enhance the full description of memory, Caputo and Fabrizio have recently introduced a new fractional order derivative without a singular kernel [16, 20, 21] . In their paper, they demonstrated that the interest in the new derivative is because of the requirement of exploiting the performance of the conventional viscoelastic materials, thermal media, electromagnetic systems and others. However, they pointed out the fact that the commonly used fractional derivatives were designed to deal with mechanical phenomena, connected to plasticity, fatigue, damage and also electromagnetic hysteresis [20] . Therefore the new derivative can be used outside the scope of the described field. More importantly, their proposed derivative is able to portray material heterogeneities and structures at different scales [20] . The aim of this paper is to check the possibility of applying this new derivative to other branches of sciences, in particular epidemiology. In this work, we will modify the model proposed by by replacing the ordinary time derivative with the Caputo-Fabrizio fractional order derivative. In the knowledge that the new derivative is not popular, we will first present some useful information about this derivative to inform those readers that are not aware of it.
doi:10.3390/e17064439 fatcat:m2tqqyq6yzehrnbcpmuqjxpzwu